I guess you mean that you want to collect/filter the set of solutions given by Solve for cases where values of both $k$ and $m$ are constant! If this is indeed the case then try the following where res is the list of $23$ solution returned by your Solve.

This gives us $10$ pairs of solutions where for a fixed value of $k$ we have same value of $m$. The Sort is used just to order the pairs according to the value of $k$. Following are the $3$ outliers that do not follow above pattern are

Mathematica is a registered trademark of Wolfram Research, Inc. While the mark is used herein with the limited permission of Wolfram Research, Stack Exchange and this site disclaim all affiliation therewith.